I have received this message which encapsulates the gist of this
discussion:
"I can only assume that you believe there is a finite number of
numbers, by precisely the same argument as you apply to natural
languages. [...] If you believe there is an infinite number of
numbers, then you contradict yourself and should rethink what you
believe about natural language."
The assumption is right. Yes, right; LITERALLY right. "Literally"
is the key: I do hold that the "number of numbers", that is, the
number of PHYSICALLY enumerable SIGNS representing numbers, is
finite. I also hold that the number of positive integers is
infinite. Ditto for the number of points on a line, or in a
plane, or in a space, only more "infinite" than the former.
Contradiction? No. Integers, geometric points have no physical
existence; linguistic signs have. There again we meet the
fundamental distinction between signifiant and signifie', sign
and referent, and between reality and abstract models of it.
Should I ramble on about what lurks behind "numbers" and
"counting"? I'd rather not. Show time, instead.
------------------------
ARTHUR: This new learning amazes me, Sir Bedevere. Explain again
how there are infinitely many ways in which Joseph could have
descended from Abraham.
BEDEVERE: Of course, my Liege. Matthew has written that Abraham
begat Isaac, who begat Jacob, who begat Juda, and so on, who
begat Joseph.
ARTHUR: Yes.
BEDEVERE: Behold then, my Liege, these generative rules:
<Genealogy of Joseph, from Abraham>
::= Abraham <more Hebrews> Joseph
<more Hebrews> ::= <one Hebrew>|
<one Hebrew><more Hebrews>
They are truly wondrous, my Liege, for they account for all
the possible filiations through which Joseph could have
descended from Abraham, including that reported by Matthew:
just replace <one Hebrew> with a Hebrew and have him
beget.
ARTHUR: Uh, yeah, I can see that. Randy lot those Hebrews, eh?
BEDEVERE: And those alternative genealogies, my Liege, are
infinite in number because <more Hebrews> is recursive.
----------------------------
As you can see, Sir Bedevere's conditions for membership of the
set of alternative genealogies include a recursive definition,
which allows him to conclude that their number is infinite.
Reinterpret "Abraham" and "Joseph" as the boundaries of a
sentence, and replace <one Hebrew> not with a Hebrew patriarch
but with a phoneme of whatever language you please. Sir
Bedevere's rules now generate all the possible sentences of that
language, and the impossible ones too.
Are there really infinitely many possible alternative ways in
which Joseph could have descended from Abraham? Yes, if there are
infinitely many sentences in any natural language. And vice
versa. For the rules are one and the same, and the cardinality of
their elements alike: there was a finite number of generations
from Abraham to Joseph, as there is a finite number of phonemes
in any sentence; there was only a finite number of Hebrews
draftable into stud duty at any time, as there is only a finite
number of phonemes you can choose from to fill any particular
position in a sentence.
>From here on, the choice is yours. But note that Sir Bedevere
does not test what his model predicts against what is physically
possible, let alone what has been physically observed. He is in
good company, with Hegel and Descartes, for whom truth was
derivable through reason alone without reference to the physical
world.
Does that settle it?

When we look at a language in its actual state, it is finite. As such
it has all the limitations of finite entities. There are, for example,
ideas that do not have a word in the lexicon to express it with
because the lexicon is finite. But, if we consider the possibility of
an open lexicon taking in new words, the infinite language, in its
abstract aspect, has the potential to express anything hitherto not
encountered. So, it is not meaningless to talk about the properties of
of a language that come with its infinite character.
I understand that the question of whether sentences can be of infinite
length is a thorny issue. I think I should not say anything definite
about it.
Tom Lai.

macrakisosf.org asks
> Assuming finite-length sentences and finite numbers of combining
> elements (whether phonemes or something else), set theory won't get
> you anywhere beyond countable infinity. How do they [Langendoen
> and Postal] get anything bigger?
They allow infinite-lenth sentences, built by closing a finite set of
finite-length sentences under conjunction. They they introduce a
Cantorian diagonalization to show the existence of new sentences not
in the original set, and thus claim to show that the size of an NL
is transfinite.
Given that the argument they present is essential no different from
Cantor's, it seems to me that if you accept the "existence" in some
sense of transfinite numbers, you can't also deny the existence of
sentence-like objects and transfinite-sized NL-like objects of the
sort Langendoen and Postal are talking about. The philosophical
problems only arise when we consider whether the NL-like objects
of transfinite size are "real languages" in the more familiar
sense. But this is the bag of worms, not L&P's mathematical
argument, it seems to me.
--- John Coleman